(- (/ x0 (- 1 x1)) x0)

Average Error: 7.9 → 6.0

Time: 8.1s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 2.985
  2. x1 = 0.000209

\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]

Math

\[\frac{x0}{1 - x1} - x0\]

↓

\[\frac{\log \left(e^{\frac{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot x0}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}\right)}{\left(x0 \cdot \sqrt[3]{\frac{x0}{1 - x1} \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right)} + x0 \cdot x0\right) + \frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}}\]

C

double f(double x0, double x1) {
        double r2913295 = x0;
        double r2913296 = 1.0;
        double r2913297 = x1;
        double r2913298 = r2913296 - r2913297;
        double r2913299 = r2913295 / r2913298;
        double r2913300 = r2913299 - r2913295;
        return r2913300;
}

↓

double f(double x0, double x1) {
        double r2913301 = x0;
        double r2913302 = 1.0;
        double r2913303 = x1;
        double r2913304 = r2913302 - r2913303;
        double r2913305 = r2913301 / r2913304;
        double r2913306 = r2913305 * r2913305;
        double r2913307 = r2913306 * r2913301;
        double r2913308 = r2913307 / r2913304;
        double r2913309 = r2913301 * r2913301;
        double r2913310 = r2913301 * r2913309;
        double r2913311 = r2913308 - r2913310;
        double r2913312 = exp(r2913311);
        double r2913313 = log(r2913312);
        double r2913314 = r2913305 * r2913306;
        double r2913315 = cbrt(r2913314);
        double r2913316 = r2913301 * r2913315;
        double r2913317 = r2913316 + r2913309;
        double r2913318 = r2913317 + r2913306;
        double r2913319 = r2913313 / r2913318;
        return r2913319;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.3
Herbie	6.0

\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

1. Initial program 7.9

   \[\frac{x0}{1 - x1} - x0\]
2. Using strategy rm

3. Applied flip3-- 7.7

   \[\leadsto \color{blue}{\frac{{\left(\frac{x0}{1 - x1}\right)}^{3} - {x0}^{3}}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \frac{x0}{1 - x1} \cdot x0\right)}}\]

4. Simplified 7.3

   \[\leadsto \frac{\color{blue}{\frac{x0}{1 - x1} \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) - x0 \cdot \left(x0 \cdot x0\right)}}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \frac{x0}{1 - x1} \cdot x0\right)}\]
5. Using strategy rm

6. Applied associate-*l/ 6.1

   \[\leadsto \frac{\color{blue}{\frac{x0 \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right)}{1 - x1}} - x0 \cdot \left(x0 \cdot x0\right)}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \frac{x0}{1 - x1} \cdot x0\right)}\]
7. Using strategy rm

8. Applied add-cbrt-cube 6.1

   \[\leadsto \frac{\frac{x0 \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right)}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \frac{x0}{\color{blue}{\sqrt[3]{\left(\left(1 - x1\right) \cdot \left(1 - x1\right)\right) \cdot \left(1 - x1\right)}}} \cdot x0\right)}\]

9. Applied add-cbrt-cube 6.1

   \[\leadsto \frac{\frac{x0 \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right)}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \frac{\color{blue}{\sqrt[3]{\left(x0 \cdot x0\right) \cdot x0}}}{\sqrt[3]{\left(\left(1 - x1\right) \cdot \left(1 - x1\right)\right) \cdot \left(1 - x1\right)}} \cdot x0\right)}\]

10. Applied cbrt-undiv 6.1

    \[\leadsto \frac{\frac{x0 \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right)}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \color{blue}{\sqrt[3]{\frac{\left(x0 \cdot x0\right) \cdot x0}{\left(\left(1 - x1\right) \cdot \left(1 - x1\right)\right) \cdot \left(1 - x1\right)}}} \cdot x0\right)}\]

11. Simplified 6.1

    \[\leadsto \frac{\frac{x0 \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right)}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \sqrt[3]{\color{blue}{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot \frac{x0}{1 - x1}}} \cdot x0\right)}\]
12. Using strategy rm

13. Applied add-log-exp 6.0

    \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{x0 \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right)}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}\right)}}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \sqrt[3]{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot \frac{x0}{1 - x1}} \cdot x0\right)}\]

14. Final simplification 6.0

    \[\leadsto \frac{\log \left(e^{\frac{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot x0}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}\right)}{\left(x0 \cdot \sqrt[3]{\frac{x0}{1 - x1} \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right)} + x0 \cdot x0\right) + \frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :pre (or (and (== x0 1.855) (== x1 0.000209)) (and (== x0 2.985) (== x1 0.0186)))

  :herbie-target
  (/ (* x0 x1) (- 1 x1))

  (- (/ x0 (- 1 x1)) x0))